Math Problem Statement

Simplify the expression \( \frac{\sqrt{3}}{\sqrt{21}} \)

Solution

The expression in the image is:

321\frac{\sqrt{3}}{\sqrt{21}}

To simplify this, we can combine the square roots under one square root as follows:

321=321=17\frac{\sqrt{3}}{\sqrt{21}} = \sqrt{\frac{3}{21}} = \sqrt{\frac{1}{7}}

Therefore, the simplified form of the expression is:

321=17\frac{\sqrt{3}}{\sqrt{21}} = \frac{1}{\sqrt{7}}

To rationalize the denominator, we multiply the numerator and denominator by 7\sqrt{7}:

17×77=77\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7}

So the final simplified expression is:

77\frac{\sqrt{7}}{7}

Would you like any further explanation or details?

Here are some related questions:

  1. How do you rationalize a denominator?
  2. What are the rules for simplifying square roots?
  3. Can square roots always be simplified in terms of integers?
  4. What is the importance of rationalizing the denominator?
  5. How would the simplification change if there was an additional term?

Tip: Always try to simplify square roots and fractions first before rationalizing the denominator to make calculations easier.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplifying Square Roots
Rationalizing Denominators

Formulas

\( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)
Rationalizing the denominator: \( \frac{1}{\sqrt{n}} = \frac{\sqrt{n}}{n} \)

Theorems

Properties of Square Roots

Suitable Grade Level

Grades 8-10